10,738 research outputs found
The real projective spaces in homotopy type theory
Homotopy type theory is a version of Martin-L\"of type theory taking
advantage of its homotopical models. In particular, we can use and construct
objects of homotopy theory and reason about them using higher inductive types.
In this article, we construct the real projective spaces, key players in
homotopy theory, as certain higher inductive types in homotopy type theory. The
classical definition of RP(n), as the quotient space identifying antipodal
points of the n-sphere, does not translate directly to homotopy type theory.
Instead, we define RP(n) by induction on n simultaneously with its tautological
bundle of 2-element sets. As the base case, we take RP(-1) to be the empty
type. In the inductive step, we take RP(n+1) to be the mapping cone of the
projection map of the tautological bundle of RP(n), and we use its universal
property and the univalence axiom to define the tautological bundle on RP(n+1).
By showing that the total space of the tautological bundle of RP(n) is the
n-sphere, we retrieve the classical description of RP(n+1) as RP(n) with an
(n+1)-cell attached to it. The infinite dimensional real projective space,
defined as the sequential colimit of the RP(n) with the canonical inclusion
maps, is equivalent to the Eilenberg-MacLane space K(Z/2Z,1), which here arises
as the subtype of the universe consisting of 2-element types. Indeed, the
infinite dimensional projective space classifies the 0-sphere bundles, which
one can think of as synthetic line bundles.
These constructions in homotopy type theory further illustrate the utility of
homotopy type theory, including the interplay of type theoretic and homotopy
theoretic ideas.Comment: 8 pages, to appear in proceedings of LICS 201
Algebraic cycles and the classical groups II: Quaternionic cycles
In part I of this work we studied the spaces of real algebraic cycles on a
complex projective space P(V), where V carries a real structure, and completely
determined their homotopy type. We also extended some functors in K-theory to
algebraic cycles, establishing a direct relationship to characteristic classes
for the classical groups, specially Stiefel-Whitney classes. In this sequel, we
establish corresponding results in the case where V has a quaternionic
structure. The determination of the homotopy type of quaternionic algebraic
cycles is more involved than in the real case, but has a similarly simple
description. The stabilized space of quaternionic algebraic cycles admits a
nontrivial infinite loop space structure yielding, in particular, a delooping
of the total Pontrjagin class map. This stabilized space is directly related to
an extended notion of quaternionic spaces and bundles (KH-theory), in analogy
with Atiyah's real spaces and KR-theory, and the characteristic classes that we
introduce for these objects are nontrivial. The paper ends with various
examples and applications.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol9/paper27.abs.htm
Twisted -theory
Twisted complex -theory can be defined for a space equipped with a
bundle of complex projective spaces, or, equivalently, with a bundle of
C-algebras. Up to equivalence, the twisting corresponds to an element of
. We give a systematic account of the definition and basic
properties of the twisted theory, emphasizing some points where it behaves
differently from ordinary -theory. (We omit, however, its relations to
classical cohomology, which we shall treat in a sequel.) We develop an
equivariant version of the theory for the action of a compact Lie group,
proving that then the twistings are classified by the equivariant cohomology
group . We also consider some basic examples of twisted -theory
classes, related to those appearing in the recent work of
Freed-Hopkins-Teleman.Comment: 49 pages;some minor corrections have been made to the earlier versio
The block structure spaces of real projective spaces and orthogonal calculus of functors
Given a compact manifold X, the set of simple manifold structures on X x
\Delta^k relative to the boundary can be viewed as the k-th homotopy group of a
space \S^s (X). This space is called the block structure space of X.
We study the block structure spaces of real projective spaces. Generalizing
Wall's join construction we show that there is a functor from the category of
finite dimensional real vector spaces with inner product to the category of
pointed spaces which sends the vector space V to the block structure space of
the projective space of V. We study this functor from the point of view of
orthogonal calculus of functors; we show that it is polynomial of degree <= 1
in the sense of orthogonal calculus.
This result suggests an attractive description of the block structure space
of the infinite dimensional real projective space via the Taylor tower of
orthogonal calculus. This space is defined as a colimit of block structure
spaces of projective spaces of finite-dimensional real vector spaces and is
closely related to some automorphisms spaces of real projective spaces.Comment: corrected version, 32 pages, published in Transactions of the AMS at
http://www.ams.org/tran/2007-359-01/S0002-9947-06-04180-8
Schematic homotopy types and non-abelian Hodge theory
In this work we use Hodge theoretic methods to study homotopy types of
complex projective manifolds with arbitrary fundamental groups. The main tool
we use is the \textit{schematization functor} , introduced by the third author as a substitute for the
rationalization functor in homotopy theory in the case of non-simply connected
spaces. Our main result is the construction of a \textit{Hodge decomposition}
on . This Hodge decomposition is encoded in an
action of the discrete group on the object
and is shown to recover the usual Hodge
decomposition on cohomology, the Hodge filtration on the pro-algebraic
fundamental group as defined by C.Simpson, and in the simply connected case,
the Hodge decomposition on the complexified homotopy groups as defined by
J.Morgan and R. Hain. This Hodge decomposition is shown to satisfy a purity
property with respect to a weight filtration, generalizing the fact that the
higher homotopy groups of a simply connected projective manifold have natural
mixed Hodge structures. As a first application we construct a new family of
examples of homotopy types which are not realizable as complex projective
manifolds. Our second application is a formality theorem for the schematization
of a complex projective manifold. Finally, we present conditions on a complex
projective manifold under which the image of the Hurewitz morphism of
is a sub-Hodge structure.Comment: 57 pages. This new version has been globally reorganized and includes
additional results and applications. Minor correction
An Abel-Jacobi invariant for cobordant cycles
We discuss an Abel-Jacobi invariant for algebraic cobordism cycles whose
image in topological cobordism vanishes. The existence of this invariant
follows by abstract arguments from the construction of Hodge filtered
cohomology theories in joint work of Michael J. Hopkins and the author. In this
paper, we give a concrete description of the new Abel-Jacobi map and Hodge
filtered cohomology groups for projective smooth complex varieties.Comment: v2: 21 pages; revised section 4; final version to appear in Doc. Mat
Topological comparison theorems for Bredon motivic cohomology
We prove equivariant versions of the Beilinson-Lichtenbaum conjecture for
Bredon motivic cohomology of smooth complex and real varieties with an action
of the group of order two. This identifies equivariant motivic and topological
invariants in a large range of degrees.Comment: Corrected indices in main theorem and a few minor changes. To appear,
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